On some special subspaces of a Banach space, from the perspective of best coapproximation

TL;DR

This paper investigates special subspaces in Banach spaces related to best coapproximation, introducing new types and providing conditions for their properties using orthogonality techniques.

Contribution

It introduces anti-coproximinal and strongly anti-coproximinal subspaces, offering necessary and sufficient conditions for their existence in various Banach space settings.

Findings

Characterized anti-coproximinal subspaces in smooth Banach spaces.

Provided conditions for strongly anti-coproximinal subspaces in reflexive Banach spaces.

Explored geometric structures of these subspaces in finite-dimensional polyhedral Banach spaces.

Abstract

We study the best coapproximation problem in Banach spaces, by using Birkhoff-James orthogonality techniques. We introduce two special types of subspaces, christened the anti-coproximinal subspaces and the strongly anti-coproximinal subspaces. We obtain a necessary condition for the strongly anti-coproximinal subspaces in a reflexive Banach space whose dual space satisfies the Kadets-Klee Property. On the other hand, we provide a sufficient condition for the strongly anti-coproximinal subspaces in a general Banach space. We also characterize the anti-coproximinal subspaces of a smooth Banach space. Further, we study these special subspaces in a finite-dimensional polyhedral Banach space and find some interesting geometric structures associated with them.